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In this paper, we discuss the properties of the embedding
operator $i^\Lambda_\mu : M_\Lambda^\infty\hookrightarrow L^\infty(\mu),$
where $\mu$ is a positive Borel measure on $[0,1]$ and $M_{\Lambda}^{\infty}$
is a Müntz space. In particular, we compute the essential norm
of this embedding. As a consequence, we recover some results
of
the first author.
We also study the compactness (resp. weak compactness)
and compute the essential norm (resp. generalized essential norm)
of the embedding $i_{\mu_1,\,\mu_2} : L^\infty(\mu_1)\hookrightarrow
L^\infty(\mu_2)$, where $\mu_1$, $\mu_2$ are two positive Borel
measures on $[0,1]$ with $\mu_2$ absolutely continuous with respect
to $\mu_1$.

In this paper, we discuss the properties of the embedding
operator $i^\Lambda_\mu : M_\Lambda^\infty\hookrightarrow L^\infty(\mu),$
where $\mu$ is a positive Borel measure on $[0,1]$ and $M_{\Lambda}^{\infty}$
is a Müntz space. In particular, we compute the essential norm
of this embedding. As a consequence, we recover some results
of
the first author.
We also study the compactness (resp. weak compactness)
and compute the essential norm (resp. generalized essential norm)
of the embedding $i_{\mu_1,\,\mu_2} : L^\infty(\mu_1)\hookrightarrow
L^\infty(\mu_2)$, where $\mu_1$, $\mu_2$ are two positive Borel
measures on $[0,1]$ with $\mu_2$ absolutely continuous with respect
to $\mu_1$.

We prove that if $C$ is a reflexive smooth plane curve of degree
$d$ defined over a finite field $\mathbb{F}_q$ with $d\leq q+1$, then
there is an $\mathbb{F}_q$-line $L$ that intersects $C$ transversely.
We also prove the same result for non-reflexive curves of degree
$p+1$ and $2p+1$ where $q=p^{r}$.

We prove that if $C$ is a reflexive smooth plane curve of degree
$d$ defined over a finite field $\mathbb{F}_q$ with $d\leq q+1$, then
there is an $\mathbb{F}_q$-line $L$ that intersects $C$ transversely.
We also prove the same result for non-reflexive curves of degree
$p+1$ and $2p+1$ where $q=p^{r}$.

According to a well-known theorem of Serre and Tate, the infinitesimal
deformation theory of an abelian variety in positive characteristic
is equivalent to the infinitesimal deformation theory of its
Barsotti-Tate group. We extend this result to $1$-motives.

According to a well-known theorem of Serre and Tate, the infinitesimal
deformation theory of an abelian variety in positive characteristic
is equivalent to the infinitesimal deformation theory of its
Barsotti-Tate group. We extend this result to $1$-motives.

Let $G$ be a claw-free graph on $n$ vertices with clique number
$\omega$, and consider the chromatic number $\chi(G^2)$ of the
square $G^2$ of $G$.
Writing $\chi'_s(d)$ for the supremum of $\chi(L^2)$ over the
line graphs $L$ of simple graphs of maximum degree at most $d$,
we prove that $\chi(G^2)\le \chi'_s(\omega)$ for $\omega \in
\{3,4\}$. For $\omega=3$, this implies the sharp bound $\chi(G^2)
\leq 10$. For $\omega=4$, this implies $\chi(G^2)\leq 22$, which
is within $2$ of the conjectured best bound.
This work is motivated by a strengthened form of a conjecture
of Erdős and Nešetřil.

Let $G$ be a claw-free graph on $n$ vertices with clique number
$\omega$, and consider the chromatic number $\chi(G^2)$ of the
square $G^2$ of $G$.
Writing $\chi'_s(d)$ for the supremum of $\chi(L^2)$ over the
line graphs $L$ of simple graphs of maximum degree at most $d$,
we prove that $\chi(G^2)\le \chi'_s(\omega)$ for $\omega \in
\{3,4\}$. For $\omega=3$, this implies the sharp bound $\chi(G^2)
\leq 10$. For $\omega=4$, this implies $\chi(G^2)\leq 22$, which
is within $2$ of the conjectured best bound.
This work is motivated by a strengthened form of a conjecture
of Erdős and Nešetřil.

In this note we give a characterization of $\ell^{p}\times \cdots\times
\ell^{p}\to\ell^q$ boundedness of maximal operators associated
to multilinear convolution averages over spheres in $\mathbb{Z}^n$.

In this note we give a characterization of $\ell^{p}\times \cdots\times
\ell^{p}\to\ell^q$ boundedness of maximal operators associated
to multilinear convolution averages over spheres in $\mathbb{Z}^n$.

We explicitly describe the isomorphism between two combinatorial
realizations of Kashiwara's infinity crystal in types B and C.
The first realization is in terms of marginally large tableaux
and the other is in terms of Kostant partitions coming from PBW
bases. We also discuss a stack notation for Kostant partitions
which simplifies that realization.

We explicitly describe the isomorphism between two combinatorial
realizations of Kashiwara's infinity crystal in types B and C.
The first realization is in terms of marginally large tableaux
and the other is in terms of Kostant partitions coming from PBW
bases. We also discuss a stack notation for Kostant partitions
which simplifies that realization.

We consider three special and significant cases of the following
problem. Let $D\subset\mathbb{R}^d$ be a (possibly unbounded) set
of finite Lebesgue measure.
Let $E( \mathbb{Z}^d)=\{e^{2\pi i x\cdot n}\}_{n\in\mathbb{Z}^d}$ be the standard
exponential basis on the unit cube of $\mathbb{R}^d$.
Find conditions on $D$ for which $E(\mathbb{Z}^d)$ is a frame, a
Riesz sequence, or a Riesz basis for $L^2(D)$.

We consider three special and significant cases of the following
problem. Let $D\subset\mathbb{R}^d$ be a (possibly unbounded) set
of finite Lebesgue measure.
Let $E( \mathbb{Z}^d)=\{e^{2\pi i x\cdot n}\}_{n\in\mathbb{Z}^d}$ be the standard
exponential basis on the unit cube of $\mathbb{R}^d$.
Find conditions on $D$ for which $E(\mathbb{Z}^d)$ is a frame, a
Riesz sequence, or a Riesz basis for $L^2(D)$.

Let $\mu$ be a positive finite Borel measure on the unit circle
and $\mathcal{D}(\mu)$ the associated harmonically weighted Dirichlet
space. In this paper we show that for each closed subset $E$
of the unit circle with zero $c_{\mu}-$capacity, there exists
a function $f\in\mathcal{D}(\mu)$ such that $f$ is cyclic (i.e.,
$\{p f: p $ is a polynomial$\}$ is dense in $\mathcal{D}(\mu)$),
$f$ vanishes on $E$, and $f$ is uniformly continuous.
Then we provide a sufficient
condition for a continuous function on the closed unit disk to
be cyclic in $\mathcal{D}(\mu)$.

Let $\mu$ be a positive finite Borel measure on the unit circle
and $\mathcal{D}(\mu)$ the associated harmonically weighted Dirichlet
space. In this paper we show that for each closed subset $E$
of the unit circle with zero $c_{\mu}-$capacity, there exists
a function $f\in\mathcal{D}(\mu)$ such that $f$ is cyclic (i.e.,
$\{p f: p $ is a polynomial$\}$ is dense in $\mathcal{D}(\mu)$),
$f$ vanishes on $E$, and $f$ is uniformly continuous.
Then we provide a sufficient
condition for a continuous function on the closed unit disk to
be cyclic in $\mathcal{D}(\mu)$.

A precise quantitative version of the following qualitative statement
is proved: If a finite dimensional normed space contains approximately
Euclidean subspaces of all proportional dimensions, then every
proportional dimensional quotient space has the same property.

A precise quantitative version of the following qualitative statement
is proved: If a finite dimensional normed space contains approximately
Euclidean subspaces of all proportional dimensions, then every
proportional dimensional quotient space has the same property.

Very recently, Karder and Petek completely described maps on density
matrices (positive semidefinite matrices with unit trace) preserving
certain entropy-like convex functionals of any convex combination.
As a result, maps could be characterized which preserve von Neumann
entropy or Schatten $p$-norm of any convex combination of quantum
states (whose mathematical representatives are the density matrices).
In this note we consider these latter two problems on the set
of invertible density operators, in a much more general setting,
on the set of positive invertible elements with unit trace in
a $C^{*}$-algebra.

Very recently, Karder and Petek completely described maps on density
matrices (positive semidefinite matrices with unit trace) preserving
certain entropy-like convex functionals of any convex combination.
As a result, maps could be characterized which preserve von Neumann
entropy or Schatten $p$-norm of any convex combination of quantum
states (whose mathematical representatives are the density matrices).
In this note we consider these latter two problems on the set
of invertible density operators, in a much more general setting,
on the set of positive invertible elements with unit trace in
a $C^{*}$-algebra.

We bring examples of toric varieties blown up at a point in the torus that do not have finitely generated Cox rings. These examples are generalizations of
our earlier work,
where toric surfaces of Picard number $1$ were studied. In this article we consider toric varieties of higher Picard number and higher dimension. In particular, we bring examples of weighted projective $3$-spaces blown up at a point that do not have finitely generated Cox rings.

We bring examples of toric varieties blown up at a point in the torus that do not have finitely generated Cox rings. These examples are generalizations of
our earlier work,
where toric surfaces of Picard number $1$ were studied. In this article we consider toric varieties of higher Picard number and higher dimension. In particular, we bring examples of weighted projective $3$-spaces blown up at a point that do not have finitely generated Cox rings.

A theorem of Gekeler compares the number of non-isomorphic automorphic
representations associated with the space of cusp forms of weight
$k$ on $\Gamma_0(N)$ to a simpler function of $k$ and $N$, showing
that the two are equal whenever $N$ is squarefree. We prove the
converse of this theorem (with one small exception), thus providing
a characterization of squarefree integers. We also establish
a similar characterization of prime numbers in terms of the number
of Hecke newforms of weight $k$ on $\Gamma_0(N)$.
It follows that a hypothetical fast algorithm for computing the
number of such automorphic representations for even a single
weight $k$ would yield a fast test for whether $N$ is squarefree.
We also show how to obtain bounds on the possible square divisors
of a number $N$ that has been found to not be squarefree via
this test, and we show how to probabilistically obtain
the complete factorization of the squarefull part of $N$ from
the number of such automorphic representations for two different
weights. If in addition we have the number of such Hecke newforms
for even a single weight $k$, then we show how to probabilistically
factor $N$ entirely.
All of these computations could be performed quickly in practice,
given the number(s) of automorphic representations and modular
forms as input.

A theorem of Gekeler compares the number of non-isomorphic automorphic
representations associated with the space of cusp forms of weight
$k$ on $\Gamma_0(N)$ to a simpler function of $k$ and $N$, showing
that the two are equal whenever $N$ is squarefree. We prove the
converse of this theorem (with one small exception), thus providing
a characterization of squarefree integers. We also establish
a similar characterization of prime numbers in terms of the number
of Hecke newforms of weight $k$ on $\Gamma_0(N)$.

It follows that a hypothetical fast algorithm for computing the
number of such automorphic representations for even a single
weight $k$ would yield a fast test for whether $N$ is squarefree.
We also show how to obtain bounds on the possible square divisors
of a number $N$ that has been found to not be squarefree via
this test, and we show how to probabilistically obtain
the complete factorization of the squarefull part of $N$ from
the number of such automorphic representations for two different
weights. If in addition we have the number of such Hecke newforms
for even a single weight $k$, then we show how to probabilistically
factor $N$ entirely.
All of these computations could be performed quickly in practice,
given the number(s) of automorphic representations and modular
forms as input.

In this short note, we prove that on the three-sphere with any
bumpy metric there exist at least two pairs of solutions of the
Allen-Cahn equation with spherical interface and index at most
two. The proof combines several recent results from the literature.

In this short note, we prove that on the three-sphere with any
bumpy metric there exist at least two pairs of solutions of the
Allen-Cahn equation with spherical interface and index at most
two. The proof combines several recent results from the literature.

In this paper we show that to a unital associative algebra object
(resp. co-unital co-associative co-algebra object) of any abelian
monoidal category $(\mathcal{C}, \otimes)$ endowed with a symmetric $2$-trace,
i.e. an $F\in Fun(\mathcal{C}, \operatorname{Vec})$ satisfying some natural trace-like
conditions, one can attach a cyclic (resp.cocyclic) module, and
therefore speak of the (co)cyclic homology of the (co)algebra
``with coefficients in $F$". Furthermore, we observe that if
$\mathcal{M}$ is a $\mathcal{C}$-bimodule category and $(F, M)$ is a stable central
pair, i.e., $F\in Fun(\mathcal{M}, \operatorname{Vec})$ and $M\in \mathcal{M}$ satisfy certain
conditions, then $\mathcal{C}$ acquires a symmetric 2-trace. The dual
notions of symmetric $2$-contratraces and stable central contrapairs
are derived as well. As an application we can recover all Hopf
cyclic type (co)homology theories.

In this paper we show that to a unital associative algebra object
(resp. co-unital co-associative co-algebra object) of any abelian
monoidal category $(\mathcal{C}, \otimes)$ endowed with a symmetric $2$-trace,
i.e. an $F\in Fun(\mathcal{C}, \operatorname{Vec})$ satisfying some natural trace-like
conditions, one can attach a cyclic (resp.cocyclic) module, and
therefore speak of the (co)cyclic homology of the (co)algebra
``with coefficients in $F$". Furthermore, we observe that if
$\mathcal{M}$ is a $\mathcal{C}$-bimodule category and $(F, M)$ is a stable central
pair, i.e., $F\in Fun(\mathcal{M}, \operatorname{Vec})$ and $M\in \mathcal{M}$ satisfy certain
conditions, then $\mathcal{C}$ acquires a symmetric 2-trace. The dual
notions of symmetric $2$-contratraces and stable central contrapairs
are derived as well. As an application we can recover all Hopf
cyclic type (co)homology theories.

Let $\beta\ge 0$ and $e_1=(1,0,\ldots,0)$ is a unit vector on
$\mathbb{R}^{n}$, $d\mu(x)=|x|^\beta dx$ is a power weighted
measure on $\mathbb{R}^n$. For $0\le \alpha\lt n$, let $M_\mu^\alpha$
be the centered Hardy-Littlewood maximal function and fractional
maximal functions associated to measure $\mu$. This paper shows
that for $q=n/(n-\alpha)$, $f\in L^1(\mathbb{R}^n,d\mu)$,
$$\lim\limits_{\lambda\to 0+}\lambda^q \mu(\{x\in\mathbb{R}^n:M_\mu^\alpha
f(x)\gt \lambda\})=\frac{\omega_{n-1}}{(n+\beta)\mu(B(e_1,1))}\|f\|_{L^1(\mathbb{R}^n,
d\mu)}^q,$$
and
$$\lim_{\lambda\to 0+}\lambda^q \mu\Big(\Big\{x\in\mathbb{R}^n:\Big|M_\mu^\alpha
f(x)-\frac{\|f\|_{L^1(\mathbb{R}^n, d\mu)}}{\mu(B(x,|x|))^{1-\alpha/n}}\Big|\gt \lambda\Big\}\Big)=0,$$
which is new and stronger than the previous result even if $\beta=0$.
Meanwhile, the corresponding results for the un-centered maximal
functions as well as the fractional integral operators with respect
to measure $\mu$ are also obtained.

Let $\beta\ge 0$ and $e_1=(1,0,\ldots,0)$ is a unit vector on
$\mathbb{R}^{n}$, $d\mu(x)=|x|^\beta dx$ is a power weighted
measure on $\mathbb{R}^n$. For $0\le \alpha\lt n$, let $M_\mu^\alpha$
be the centered Hardy-Littlewood maximal function and fractional
maximal functions associated to measure $\mu$. This paper shows
that for $q=n/(n-\alpha)$, $f\in L^1(\mathbb{R}^n,d\mu)$,
$$\lim\limits_{\lambda\to 0+}\lambda^q \mu(\{x\in\mathbb{R}^n:M_\mu^\alpha
f(x)\gt \lambda\})=\frac{\omega_{n-1}}{(n+\beta)\mu(B(e_1,1))}\|f\|_{L^1(\mathbb{R}^n,
d\mu)}^q,$$
and
$$\lim_{\lambda\to 0+}\lambda^q \mu\Big(\Big\{x\in\mathbb{R}^n:\Big|M_\mu^\alpha
f(x)-\frac{\|f\|_{L^1(\mathbb{R}^n, d\mu)}}{\mu(B(x,|x|))^{1-\alpha/n}}\Big|\gt \lambda\Big\}\Big)=0,$$
which is new and stronger than the previous result even if $\beta=0$.
Meanwhile, the corresponding results for the un-centered maximal
functions as well as the fractional integral operators with respect
to measure $\mu$ are also obtained.

In this paper, we completely characterize the finite rank commutator
and semi-commutator of two
monomial-type Toeplitz operators on the Bergman space of certain
weakly pseudoconvex domains.
Somewhat surprisingly, there are not only plenty of commuting
monomial-type Toeplitz operators but also non-trivial semi-commuting
monomial-type Toeplitz operators. Our results are new even for
the unit ball.

In this paper, we completely characterize the finite rank commutator
and semi-commutator of two
monomial-type Toeplitz operators on the Bergman space of certain
weakly pseudoconvex domains.
Somewhat surprisingly, there are not only plenty of commuting
monomial-type Toeplitz operators but also non-trivial semi-commuting
monomial-type Toeplitz operators. Our results are new even for
the unit ball.

We find an explicit expression for the zeta-regularized determinant
of (the Friedrichs extensions) of the Laplacians on a compact
Riemann surface of genus one with conformal metric of curvature
$1$ having a single conical singularity of angle $4\pi$.

We find an explicit expression for the zeta-regularized determinant
of (the Friedrichs extensions) of the Laplacians on a compact
Riemann surface of genus one with conformal metric of curvature
$1$ having a single conical singularity of angle $4\pi$.

We prove that if $\mathfrak{s}$ is a solvable Lie algebra of matrices
over a field of characteristic 0,
and $A\in\mathfrak{s}$,
then the semisimple and nilpotent summands of the Jordan-Chevalley
decomposition of $A$ belong to $\mathfrak{s}$
if and only if there exist $S,N\in\mathfrak{s}$, $S$ is semisimple, $N$
is nilpotent (not necessarily $[S,N]=0$)
such that $A=S+N$.

We prove that if $\mathfrak{s}$ is a solvable Lie algebra of matrices
over a field of characteristic 0,
and $A\in\mathfrak{s}$,
then the semisimple and nilpotent summands of the Jordan-Chevalley
decomposition of $A$ belong to $\mathfrak{s}$
if and only if there exist $S,N\in\mathfrak{s}$, $S$ is semisimple, $N$
is nilpotent (not necessarily $[S,N]=0$)
such that $A=S+N$.

Darmon, Lauder and Rotger conjectured that the relative tangent space of the eigencurve at a classical, ordinary, irregular weight one point is of dimension two. This space can be identified with the space of normalized overconvergent generalized eigenforms, whose Fourier coefficients can be conjecturally described explicitly in terms of $p$-adic logarithms of algebraic numbers. This article presents the proof of this conjecture in the case where the weight one point is the intersection of two Hida families of Hecke theta series.

Darmon, Lauder and Rotger conjectured that the relative tangent space of the eigencurve at a classical, ordinary, irregular weight one point is of dimension two. This space can be identified with the space of normalized overconvergent generalized eigenforms, whose Fourier coefficients can be conjecturally described explicitly in terms of $p$-adic logarithms of algebraic numbers. This article presents the proof of this conjecture in the case where the weight one point is the intersection of two Hida families of Hecke theta series.

In this paper, we study the warped structures of Finsler metrics.
We obtain the differential equation that characterizes the Finsler
warped product metrics with vanishing Douglas curvature. By solving
this equation, we obtain all Finsler warped product Douglas metrics.
Some new Douglas Finsler metrics of this type are produced by
using known spherically symmetric Douglas metrics.

In this paper, we study the warped structures of Finsler metrics.
We obtain the differential equation that characterizes the Finsler
warped product metrics with vanishing Douglas curvature. By solving
this equation, we obtain all Finsler warped product Douglas metrics.
Some new Douglas Finsler metrics of this type are produced by
using known spherically symmetric Douglas metrics.

We introduce the concept of
$\{\sigma , \tau \}$-Rota-Baxter operator, as a twisted version
of a Rota-Baxter operator of weight zero. We show how to
obtain a certain $\{\sigma , \tau \}$-Rota-Baxter operator from
a solution of the associative (Bi)Hom-Yang-Baxter equation, and,
in a compatible way,
a Hom-pre-Lie algebra from an infinitesimal Hom-bialgebra.

We introduce the concept of
$\{\sigma , \tau \}$-Rota-Baxter operator, as a twisted version
of a Rota-Baxter operator of weight zero. We show how to
obtain a certain $\{\sigma , \tau \}$-Rota-Baxter operator from
a solution of the associative (Bi)Hom-Yang-Baxter equation, and,
in a compatible way,
a Hom-pre-Lie algebra from an infinitesimal Hom-bialgebra.

Let $A$ be the inductive limit of a sequence
$$
A_1\,\xrightarrow{\phi_{1,2}}\,A_2\,\xrightarrow{\phi_{2,3}}\,A_3\rightarrow\cdots
$$
with $A_n=\bigoplus_{i=1}^{n_i}A_{[n,i]}$, where all the $A_{[n,i]}$
are Elliott-Thomsen algebras and $\phi_{n,n+1}$ are homomorphisms.
In this paper, we will prove that $A$ can be written as another
inductive limit
$$
B_1\,\xrightarrow{\psi_{1,2}}\,B_2\,\xrightarrow{\psi_{2,3}}\,B_3\rightarrow\cdots
$$
with $B_n=\bigoplus_{i=1}^{n_i'}B_{[n,i]'}$, where all the $B_{[n,i]'}$
are Elliott-Thomsen algebras and with the extra condition that
all the $\psi_{n,n+1}$ are injective.

Let $A$ be the inductive limit of a sequence
$$
A_1\,\xrightarrow{\phi_{1,2}}\,A_2\,\xrightarrow{\phi_{2,3}}\,A_3\rightarrow\cdots
$$
with $A_n=\bigoplus_{i=1}^{n_i}A_{[n,i]}$, where all the $A_{[n,i]}$
are Elliott-Thomsen algebras and $\phi_{n,n+1}$ are homomorphisms.
In this paper, we will prove that $A$ can be written as another
inductive limit
$$
B_1\,\xrightarrow{\psi_{1,2}}\,B_2\,\xrightarrow{\psi_{2,3}}\,B_3\rightarrow\cdots
$$
with $B_n=\bigoplus_{i=1}^{n_i'}B_{[n,i]'}$, where all the $B_{[n,i]'}$
are Elliott-Thomsen algebras and with the extra condition that
all the $\psi_{n,n+1}$ are injective.

An odd Fredholm module for a given invertible operator on a Hilbert
space is specified by an unbounded so-called Dirac operator with
compact resolvent and bounded commutator with the given invertible.
Associated to this is an index pairing in terms of a Fredholm
operator with Noether index. Here it is shown by a spectral flow
argument how this index can be calculated as the signature of
a finite dimensional matrix called the spectral localizer.

An odd Fredholm module for a given invertible operator on a Hilbert
space is specified by an unbounded so-called Dirac operator with
compact resolvent and bounded commutator with the given invertible.
Associated to this is an index pairing in terms of a Fredholm
operator with Noether index. Here it is shown by a spectral flow
argument how this index can be calculated as the signature of
a finite dimensional matrix called the spectral localizer.

We first provide a necessary and sufficient condition for a ruled
real hypersurface in a nonflat complex space form to have constant
mean curvature in terms of integral curves of the characteristic
vector field on it.
This yields a characterization of minimal ruled real hypersurfaces
by circles.
We next characterize the homogeneous minimal ruled real hypersurface
in a complex hyperbolic space by using the notion of strongly
congruency of curves.

We first provide a necessary and sufficient condition for a ruled
real hypersurface in a nonflat complex space form to have constant
mean curvature in terms of integral curves of the characteristic
vector field on it.
This yields a characterization of minimal ruled real hypersurfaces
by circles.
We next characterize the homogeneous minimal ruled real hypersurface
in a complex hyperbolic space by using the notion of strongly
congruency of curves.

Katz and Sarnak predicted that the one level density of the zeros
of a family of $L$-functions would fall into one of five categories.
In this paper, we show that the one level density for $L$-functions
attached to cubic Galois number fields falls into the category
associated with unitary matrices.

Katz and Sarnak predicted that the one level density of the zeros
of a family of $L$-functions would fall into one of five categories.
In this paper, we show that the one level density for $L$-functions
attached to cubic Galois number fields falls into the category
associated with unitary matrices.

Free binary systems are shown not to admit idempotent means.
This refutes a conjecture of the author.
It is also shown that the extension of Hindman's theorem to
nonassociative binary systems formulated and conjectured
by the author
is false.

Free binary systems are shown not to admit idempotent means.
This refutes a conjecture of the author.
It is also shown that the extension of Hindman's theorem to
nonassociative binary systems formulated and conjectured
by the author
is false.

show that Hermite's theorem fails for every
integer $n$ of the form $3^{k_1}+3^{k_2}+3^{k_3}$
with integers $k_1\gt k_2\gt k_3\geq 0$. This confirms
a conjecture of Brassil and Reichstein. We also
obtain new results for the relative
Hermite-Joubert problem over a finitely generated
field of characteristic $0$.

show that Hermite's theorem fails for every
integer $n$ of the form $3^{k_1}+3^{k_2}+3^{k_3}$
with integers $k_1\gt k_2\gt k_3\geq 0$. This confirms
a conjecture of Brassil and Reichstein. We also
obtain new results for the relative
Hermite-Joubert problem over a finitely generated
field of characteristic $0$.

The edge-of-the-wedge theorem in several complex variables gives
the analytic continuation of functions defined on the poly upper
half plane and the poly lower half plane, the set of points in
$\mathbb{C}^{n}$ with all coordinates in the upper and lower
half planes respectively, through a set in real space, $\mathbb{R}^{n}$.
The geometry of the set in the real space can force the function
to analytically continue within the boundary itself, which is
qualified in our wedge-of-the-edge theorem. For example, if a
function extends to the union of two cubes in $\mathbb{R}^{n}$
that are positively oriented with some small overlap, the
functions must analytically continue to a neighborhood of that
overlap of a fixed size not depending of the size of the overlap.

The edge-of-the-wedge theorem in several complex variables gives
the analytic continuation of functions defined on the poly upper
half plane and the poly lower half plane, the set of points in
$\mathbb{C}^{n}$ with all coordinates in the upper and lower
half planes respectively, through a set in real space, $\mathbb{R}^{n}$.
The geometry of the set in the real space can force the function
to analytically continue within the boundary itself, which is
qualified in our wedge-of-the-edge theorem. For example, if a
function extends to the union of two cubes in $\mathbb{R}^{n}$
that are positively oriented with some small overlap, the
functions must analytically continue to a neighborhood of that
overlap of a fixed size not depending of the size of the overlap.

We prove a function field analogue of Maynard's celebrated result
about primes with restricted digits. That is, for certain ranges
of parameters $n$ and $q$, we prove an asymptotic formula for
the number of irreducible polynomials of degree $n$ over a finite
field $\mathbb{F}_q$ whose coefficients are restricted to lie
in a given subset of $\mathbb{F}_q$

We prove a function field analogue of Maynard's celebrated result
about primes with restricted digits. That is, for certain ranges
of parameters $n$ and $q$, we prove an asymptotic formula for
the number of irreducible polynomials of degree $n$ over a finite
field $\mathbb{F}_q$ whose coefficients are restricted to lie
in a given subset of $\mathbb{F}_q$

F. Cukierman asked whether or not for every
smooth
real plane curve $X \subset \mathbb{P}^2$ of even degree $d \geqslant
2$
there exists a real line
$L \subset \mathbb{P}^2$ such $X \cap L$ has no real points.
We show that the answer is ``yes" if $d = 2$ or $4$ and ``no"
if $n \geqslant 6$.

F. Cukierman asked whether or not for every
smooth
real plane curve $X \subset \mathbb{P}^2$ of even degree $d \geqslant
2$
there exists a real line
$L \subset \mathbb{P}^2$ such $X \cap L$ has no real points.
We show that the answer is ``yes" if $d = 2$ or $4$ and ``no"
if $n \geqslant 6$.

We give a Hopf boundary point lemma for weak solutions of linear
divergence form uniformly elliptic equations, with Hölder
continuous top-order coefficients and lower-order coefficients
in a Morrey space.

We give a Hopf boundary point lemma for weak solutions of linear
divergence form uniformly elliptic equations, with Hölder
continuous top-order coefficients and lower-order coefficients
in a Morrey space.

This paper provides short proofs of two fundamental theorems
of finite semigroup theory whose previous proofs were significantly
longer, namely the two-sided Krohn-Rhodes decomposition theorem
and Henckell's aperiodic pointlike theorem, using a new algebraic
technique that we call the merge decomposition. A prototypical
application of this technique decomposes a semigroup $T$ into
a two-sided semidirect product whose components are built from
two subsemigroups $T_1,T_2$, which together generate $T$, and
the subsemigroup generated by their setwise product $T_1T_2$.
In this sense we decompose $T$ by merging the subsemigroups
$T_1$ and $T_2$. More generally, our technique merges semigroup
homomorphisms from free semigroups.

This paper provides short proofs of two fundamental theorems
of finite semigroup theory whose previous proofs were significantly
longer, namely the two-sided Krohn-Rhodes decomposition theorem
and Henckell's aperiodic pointlike theorem, using a new algebraic
technique that we call the merge decomposition. A prototypical
application of this technique decomposes a semigroup $T$ into
a two-sided semidirect product whose components are built from
two subsemigroups $T_1,T_2$, which together generate $T$, and
the subsemigroup generated by their setwise product $T_1T_2$.
In this sense we decompose $T$ by merging the subsemigroups
$T_1$ and $T_2$. More generally, our technique merges semigroup
homomorphisms from free semigroups.

Suppose that $D\subset\mathbb{C}$ is a simply connected subdomain
containing the origin and $f(z_1)$ is a normalized convex (resp.,
starlike) function on $D$. Let
$$
\Omega_{N}(D)=\{(z_1,w_1,\ldots,w_k)\in \mathbb{C}\times{\mathbb{C}}^{n_1}\times\cdots\times{\mathbb{C}}^{n_k}:
\|w_1\|_{p_1}^{p_1}+\cdots+\|w_k\|_{p_k}^{p_k}<\frac{1}{\lambda_{D}(z_1)}\},$$
where $p_j\geq 1$, $N=1+n_1+\cdots+n_k,\,w_1\in{\mathbb{C}}^{n_1},\ldots,w_k\in{\mathbb{C}}^{n_k}$
and $\lambda_{D}$ is the density of the hyperbolic metric on
$D$. In this paper, we prove that
\begin{equation*}
\Phi_{N,{1/p_{1}},\cdots,{1}/{p_{k}}}(f)(z_1,w_1,\ldots,w_k)=\big(f(z_{1}),
(f'(z_{1}))^{1/p_{1}}w_1,\cdots,(f'(z_{1}))^{1/p_{k}}w_k\big)
\end{equation*}
is a normalized convex (resp., starlike) mapping on $\Omega_{N}(D)$.
If $D$ is the unit disk, then our result reduces to Gong and
Liu \cite{GL2} via a new method. Moreover, we give a new operator
for
convex mapping construction on an unbounded domain in ${\mathbb{C}}^{2}$.
By using a geometric approach, we prove that $\Phi_{N,{1/p_{1}},\cdots,{1}/{p_{k}}}(f)$
is a spirallike mapping of type $\alpha$ when $f$ is
a spirallike function of type $\alpha$ on the unit disk.

Let $\mathcal{D}$ be the irreducible Hermitian symmetric domain
of type $D_{2n}^{\mathbb{H}}$. There exists a canonical Hermitian
variation of real Hodge structure $\mathcal{V}_{\mathbb{R}}$
of Calabi-Yau type over $\mathcal{D}$. This short note concerns
the problem of giving motivic realizations for $\mathcal{V}_{\mathbb{R}}$.
Namely, we specify a descent of $\mathcal{V}_{\mathbb{R}}$ from
$\mathbb{R}$ to $\mathbb{Q}$ and ask whether the $\mathbb{Q}$-descent
of $\mathcal{V}_{\mathbb{R}}$ can be realized as sub-variation
of rational Hodge structure of those coming from families of
algebraic varieties. When $n=2$, we give a motivic realization
for $\mathcal{V}_{\mathbb{R}}$. When $n \geq 3$, we show that
the unique irreducible factor of Calabi-Yau type in $\mathrm{Sym}^2
\mathcal{V}_{\mathbb{R}}$ can be realized motivically.

Let $\mathcal{D}$ be the irreducible Hermitian symmetric domain
of type $D_{2n}^{\mathbb{H}}$. There exists a canonical Hermitian
variation of real Hodge structure $\mathcal{V}_{\mathbb{R}}$
of Calabi-Yau type over $\mathcal{D}$. This short note concerns
the problem of giving motivic realizations for $\mathcal{V}_{\mathbb{R}}$.
Namely, we specify a descent of $\mathcal{V}_{\mathbb{R}}$ from
$\mathbb{R}$ to $\mathbb{Q}$ and ask whether the $\mathbb{Q}$-descent
of $\mathcal{V}_{\mathbb{R}}$ can be realized as sub-variation
of rational Hodge structure of those coming from families of
algebraic varieties. When $n=2$, we give a motivic realization
for $\mathcal{V}_{\mathbb{R}}$. When $n \geq 3$, we show that
the unique irreducible factor of Calabi-Yau type in $\mathrm{Sym}^2
\mathcal{V}_{\mathbb{R}}$ can be realized motivically.